I have no hands on with a Famicom.

The only time I've seen this issue come up was with SNES's and ColecoVisions. 'Sometimes' the problem was caused by the RF shield (internal sheet metal shielding) shorting something out. In these instances it was remedied by removing the not necessarily necessary RF shielding. Re-seating it could help too.

It's an easy thing to try, with little risk of doing any damage.

Good Luck!